On a nonlinear hyperbolic equation with a bistable reaction term

Nonlinear second-order hyperbolic equations are gaining ground as models in many areas of application, as extensions of parabolic reaction–diffusion equations that might otherwise be used. The theory of travelling-wave solutions of such reaction–diffusion equations is well established. The present paper is concerned with its counterpart for the wider class of equations in the particular case that the reaction term is bistable. Conditions that are necessary and sufficient for the existence and uniqueness of these solutions are determined. A combination of traditional ordinary differential equation techniques and an innovatory integral equation approach is employed.